reserve i,j,k for Nat,
  r,s,r1,r2,s1,s2,sb,tb for Real,
  x for set,
  GX for non empty TopSpace;

theorem
  for A,B,C being Subset of GX st C is a_component & A c= C & B is
  connected & Cl A meets Cl B holds B c= C
proof
  let A,B,C be Subset of GX;
  assume that
A1: C is a_component and
A2: A c= C and
A3: B is connected and
A4: (Cl A) /\ (Cl B) <>{};
  consider p being Point of GX such that
A5: p in (Cl A) /\ (Cl B) by A4,SUBSET_1:4;
  reconsider C9 = C as Subset of GX;
  C9 is closed by A1,CONNSP_1:33;
  then Cl C = C by PRE_TOPC:22;
  then
A6: Cl A c= C by A2,PRE_TOPC:19;
  p in (Cl A) by A5,XBOOLE_0:def 4;
  then
A7: Component_of p=C9 by A1,A6,CONNSP_1:41;
  p in (Cl B) by A5,XBOOLE_0:def 4;
  then
A8: Cl B c= Component_of p by A3,Th1,CONNSP_1:19;
  B c= Cl B by PRE_TOPC:18;
  hence thesis by A7,A8;
end;
